Where the number in each "difference" column is the difference between two successive numbers in the previous column. Of course, since the y value itself is gotten by adding squares, the first difference is just the square itself. The second differences are just odd numbers and the third differences are the same: 2. All succeeding differences are 0.

Newton's polynomial now is y+ (dy)n+ (d2y/2) n(n-1)+ (d3y/3!)n(n-1)(n-2)+ ... with the ith term being (diy)/n! n(n-1)..(n-i+1)

so the terms in / / are going to be repeated again, and you divide that by two so you get the series below, and you get (102)(101)(100)/24 as the sum which is (2n+2)(2n+1)(2n)/24 which is equal to the sum of the nth squares.
kind of a cool way. to find the sum of cubes, use four digit numbers.